Answers:
These are the statements that apply:
The initial value is 3.
The range is y >0.
The simplified base is 8.
Explanation:
1) Given expression:
[tex] f(x)=3(16)^{\frac{3}{4} x [/tex]
2) Check every statement:
a) The initial value is 3?
initial value ⇒ x = 0 ⇒
[tex] f(0)=3(16)^{0}=3(1)=3 [/tex]
∴ The statement is right.
b) The initial value is 48?
Not, as it was already proved that it is 3.
c) The domain is x > 0?
No, because the domain of the exponential functions is all the Real numbers.
d) The range is y > 0?
That is correct, the exponential function is continuous, and monotonon increasing.
The limit when x → - ∞ is zero, but y never reaches zero, and the limit when x → ∞ is + ∞, meaning that the range is y > 0.
e) The simplified base is 12?
This is how you simplify the base:
[tex] 3(16)^{\frac{3}{4} x}=3{{(16}^{(3/4)})}^x=3(16^{3/4}})^{x}=3((2^4)^{3/4})^x=3(2^3)^x=3(8)^x [/tex]
Which shows that the simplified base is 8 (not 12).
f) The simplified base is 8?
Yes; this was just proved.
